Uniform distribution of zeroes of $L$-functions of modular forms

TL;DR

This paper proves that, assuming GRH, the zeros of L-functions associated with modular forms become uniformly distributed along the critical line as the level and weight grow large.

Contribution

It establishes the uniform distribution of zeros of L-functions of modular forms under GRH as level and weight tend to infinity.

Findings

Zeros of L-functions become uniformly distributed on the critical line

Distribution result holds under GRH assumption

Distribution improves with increasing level and weight

Abstract

We prove under GRH that zeros of $L$-functions of modular forms of level $N$ and weight $k$ become uniformly distributed on the critical line when $N+k\to\infty.$